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I would like to know some references regarding $C^{*}$ and $W^{*}$-algebras and quantum theories.

I'm interested in concrete physical applications, models and problems.

Here it is the list of references I already know:

  • Dixmier: $C^{*}$-algebras

  • Dixmier: $W^{*}$-algebras

  • Pedersen: $C^{*}$-algebras and their automorphic groups

  • Landsman: Lecture notes on $C^{*}$-algebras and quantum Mechanics

  • Araki: The mathematical theory of quantum fields

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Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

Related: and links therein. – Qmechanic Apr 10 '14 at 15:20
see also the answers at , among others mine. – Arnold Neumaier Feb 11 at 20:00

If you are interested in physical applications you could also include:

Bratteli-Robinson: Operator algebras and quantum statistical mechanics

It is a two-volume quite complete book, mathematically minded, discussing lots of applications of operator algebras theory to several physical systems, especially arising from statistical mechanics.

Haag: Local quantum physics

It is an, in my opinion, important book on modern mathematical physics (although very often mathematical proofs are only sketched) discussing the local operator algebras formulation of quantum theories, especially, quantum field theory (relying on the well known Haag-Kastler theory). The second edition is considerably better than the first one.

Sewell: Quantum Mechanics and its emergent macrophysics

It is a relatively recent book containing several applications of operators algebras especially to quantum statistical mechanics. The style is less mathematical than the one of the previous pair of books.

As general references, in addition to those you already mentioned, I also suggest the classical mathematical books on the subject:

Kadison-Ringrose: Fundamentals of the theory of Operators Algebras

Takesaki: Theory of Operator Algebras

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Thank You. In any case i did not mention that I'm interested particularly in applications in gauge theories. – Ilcapitano Apr 10 '14 at 15:15

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